P(B and A) = P(A) * P(B|A)

To make it clearer, shouldn't this step be = P(B) * P(A|B)?

Also, are middle school pupil in the U.S. familiar with the notation? Maybe one should state it in English instead?

The probability of A if B is known to occur is equal to the probability of A times the probability of B if A is known to occur divided by the probability of B.

ETA

P(A and B) = P(A) * P(B|A)

The probability of A and B to occur at the same time is the same as the probability of A to occur alone times the probability of B to occur under the condition that A is known to occur.